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Stochastic simulation of reaction-diffusion systems: A fluctuating-hydrodynamics approach.
Changho Kim1, Andy Nonaka1, John B Bell1
1Computational Research Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA.
The Journal of Chemical Physics
|April 8, 2017
Summary
We developed efficient numerical methods for stochastic reaction-diffusion systems using fluctuating hydrodynamics. These methods accurately capture thermodynamic fluctuations, accelerating pattern formation and influencing chemical front propagation.
Area of Science:
- Computational chemistry
- Chemical physics
- Biophysics
Background:
- Stochastic reaction-diffusion systems are crucial for modeling phenomena with low molecule counts.
- Existing methods like the reaction-diffusion master equation (RDME) become computationally expensive as molecule numbers increase.
- Fluctuating hydrodynamics (FHD) offers a promising alternative for simulating these systems across various fluctuation regimes.
Purpose of the Study:
- To develop novel, efficient numerical methods for stochastic reaction-diffusion systems.
- To extend the applicability of fluctuating hydrodynamics (FHD) to reaction-diffusion processes.
- To investigate the impact of thermodynamic fluctuations on pattern formation and chemical front propagation.
Main Methods:
- Formulated stochastic partial differential equations (SPDEs) for reaction-diffusion systems based on FHD.
- Employed implicit treatment of diffusion to overcome time step limitations.
- Developed predictor-corrector schemes using stochastic Crank-Nicolson for diffusion and Gillespie or tau leaping for reactions.
- Utilized an implicit midpoint tau leaping scheme for second-order weak accuracy.
Main Results:
- The FHD-based methods naturally bridge the gap between strong fluctuation regimes and the deterministic limit.
- Implicit diffusion treatment enabled significantly larger time step sizes compared to explicit methods.
- An implicit midpoint tau leaping scheme achieved second-order weak accuracy and stable structure factors.
- Simulations of the Schlögl model demonstrated fluctuations accelerate pattern formation and alter chemical front propagation.
Conclusions:
- FHD-based numerical methods provide an efficient and accurate alternative to RDME for stochastic reaction-diffusion systems.
- Thermodynamic fluctuations play a critical role in pattern formation (e.g., Turing-like patterns) and chemical wave dynamics.
- These methods offer a powerful tool for studying complex phenomena in chemistry and biology where fluctuations are significant.